The massive increase in demand for high bandwidth for network based applications such as cloud computing, video on demand, tele-presence and the like has led to an immediate need for increasing data rates for current backbone transmission networks such as those provided by optical fiber links. Data rates can be increased by increasing the spectral efficiency i.e. bits per second per unit bandwidth (Hertz) of the existing fiber optic link. A potential solution to the increasing spectral efficiency requirements for optical fiber communications is the use of non-orthogonal transmission through FTN signaling. FTN signaling is a linear modulation scheme that improves the spectral efficiency by reducing the time and/or frequency spacing between two adjacent pulses, thus introducing inter-symbol interference (ISI) and/or inter-carrier interference (ICI). Alternatively, FTN is a technique that allows for increased bit rate while preserving the signaling bandwidth by sending the data bearing pulses faster than what is recommended by “Nyquist's criterion” for ISI-free transmission. If it can be ensured that the ISI introduced by FTN transmission can be adequately compensated for, a higher transmission rate is possible with only nominal increase in the signal-to-noise ratio (SNR) of the signal at the cost of relatively higher receiver complexity.
Furthermore, modern high-performance communication systems frequently employ sophisticated forward error correction (FEC) codes such as turbo codes, low-density parity-check (LDPC) codes etc., to lower the overall bit error rate (BER). When FTN is used in conjunction with FECs, soft Viterbi algorithm (SOYA) based maximum-likelihood sequence estimation (MLSE) and Bahl-Cocke-Jelinek-Raviv (BOR) algorithm based maximum a posteriori (MAP) symbol-probability methods are considered to be the practical close-to-optimal approaches for FTN equalization to produce inputs to the FEC decoder. See for example “Receivers for Faster-than-Nyquist Signaling with and without Turbo Equalization”, A Prlja, J. B. Anderson and F. Rusek, IEEE Int. Symp. on Inf. Theory, 2008, “Reduced-complexity Receivers for Strongly Narrowband Intersymbol Interference Introduce by Faster-than-Nyquist Signaling”, A. Prlja and J. B. Anderson, IEEE Trans. Commun., 2012, and “High Order Modulation in Faster-than-Nyquist Signaling Communication Systems”, J. Yu, J. Park, F. Rusek, B. Kudrayashov and I. Bocharova, IEEE 80th Veh. Tech. Conf. Fall (VTC) 2014.
The computational complexity of the above equalization schemes can be extremely high. Among sub-optimal low-complexity receivers, linear equalizers and decision-feedback equalizers (DFEs) are potential candidates, which however suffer from performance degradation and in case of DFE, error propagation is a known major issue.
Tomlinson-Harashima Precoding (THP) has been used in systems to pre-compensate ISI introduced by the channel. However, conventional applications of THP at the transmitter rely on feedback of the channel information from the receiver to estimate the ISI introduced by the channel. For more details of conventional applications of THP, see, for example M. Tomlinson, “New Automatic Equalizer Employing Modulo Arithmetic”, Electronics Letters, 1971, and H. Harashima and H. Miyakawa, “Matched-Transmission Technique for Channels with Intersymbol Interference”, IEEE Trans. Commun., 1972.
The application of THP includes a modulo-operation at the transmitter and at the receiver and it is known to suffer from an associated “modulo loss”. A modulo operation of the THP receiver in systems that pre-compensate for ISI introduced by the channel keeps the received signal inside the modulo-boundary [M, M) for an M-ary pulse amplitude modulation (PAM) symbol (or equivalently an M2-ary quadrature amplitude modulation (QAM) symbol) which might cause (in the low to moderate signal-to-noise ratio (SNR) regime) the received symbols to be wrapped around to the wrong side of the constellation. This, in turn, may give rise to erroneous log-likelihood ratio (LLR) computation, which is required by the FEC (e.g. LDPC) decoder as intrinsic information. These inaccurate LLR values can cause significant degradation in the bit-error-rate (BER) performance. For example, in coherent optical systems where the FEC input is required to reach a certain threshold BER in order to ensure error-free transmission at the output of the LLR, the condition may no longer hold true due to the erroneous LLR distribution caused by the modulo loss.